Advanced Chemistry - Mono Mole

August 29, 2024October 7, 2024

Binomial theorem and the binomial series

The binomial theorem (or binomial expansion) expresses how to expand the power of a sum of two variables into a series of terms involving the individual powers of each variable, with coefficients determined by their positions in the expansion.

In general, the binomial expansion of $(x+y)^n$ is

$(x+y)^n=\sum_{k=0}^n\begin{pmatrix}n\\k\end{pmatrix}x^ky^{n-k}\;\;\;\;\;\;\;\;319$

where $x,y$ are variables, $n$ and $k$ are non-negative integers and $\begin{pmatrix}n\\k\end{pmatrix}=\frac{n!}{k!(n-k)!}$ are the binomial coefficients.

The binomial theorem can be proven by induction. For $n=0$ , we have

$(x+y)^0=1=\begin{pmatrix}0\\0\end{pmatrix}x^0y^0$

If the theorem holds for all $k$ when $n=m$ , then for $n=m+1$ ,

$\begin{align}(x+y)^{m+1}&=(x+y)\sum_{k=0}^m\begin{pmatrix}m\\k\end{pmatrix}x^ky^{m-k}\\&=\sum_{k=0}^m\begin{pmatrix}m\\k\end{pmatrix}x^{k+1}y^{m-k}+\sum_{k=0}^m\begin{pmatrix}m\\k\end{pmatrix}x^ky^{m-k+1}\\&=\sum_{k=0}^{m-1}\begin{pmatrix}m\\k\end{pmatrix}x^{k+1}y^{m-k}+\begin{pmatrix}m\\m\end{pmatrix}x^{m+1}+\begin{pmatrix}m\\0\end{pmatrix}y^{m+1}+\sum_{k=1}^{m}\begin{pmatrix}m\\k\end{pmatrix}x^{k}y^{m-k+1}\end{align}$

Substituting $k=u-1$ for the first summation yields $\sum_{u=1}^m\begin{pmatrix}m\\u-1\end{pmatrix}x^uy^{m-u+1}$ . Changing the index from $u$ back to $k$ gives

$\begin{align}(x+y)^{m+1}&=\sum_{k=1}^m\begin{pmatrix}m\\k-1\end{pmatrix}x^ky^{m-k+1}+x^{m+1}+y^{m+1}+\sum_{k=1}^m\begin{pmatrix}m\\k\end{pmatrix}x^ky^{m-k+1}\\&=y^{m+1}+\sum_{k=1}^m\biggr\[\begin{pmatrix}m\\k-1\end{pmatrix}+\begin{pmatrix}m\\k\end{pmatrix}\biggr\]x^ky^{m-k+1}+x^{m+1}\end{align}$

Question

Show that $\begin{pmatrix}m\\k-1\end{pmatrix}+\begin{pmatrix}m\\k\end{pmatrix}=\begin{pmatrix}m+1\\k\end{pmatrix}$ .

Answer

$\begin{align}\begin{pmatrix}m\\k-1\end{pmatrix}+\begin{pmatrix}m\\k\end{pmatrix}&=\frac{m!}{(k-1)!(m-k+1)!}+\frac{m!}{k!(m-k)!}\\&=\frac{m!}{(k-1)!(m-k+1)(m-k)!}+\frac{m!}{k(k-1)!(m-k)!}\\&=\frac{m!k+m!(m-k+1)}{k(m-k+1)(k-1)!(m-k)!}\\&=\frac{m!(m+1)}{k(m-k+1)(k-1)!(m-k)!}\\&=\frac{(m+1)!}{k!(m-k+1)!}\\&=\begin{pmatrix}m+1\\k\end{pmatrix}\end{align}$

Therefore,

$(x+y)^{m+1}=y^{m+1}+\sum_{k=1}^m\begin{pmatrix}m+1\\k\end{pmatrix}x^ky^{m-k+1}+x^{m+1}=\sum_{k=0}^{m+1}\begin{pmatrix}m+1\\k\end{pmatrix}x^ky^{m-k+1}$

and the theorem holds for all $n$ and $k$ .

The binomial theorem defines $0^0=1$ . This is evident from eq319, where

$1=(0+1)^2=\begin{pmatrix}2\\0\end{pmatrix}0^01^2+\begin{pmatrix}2\\1\end{pmatrix}0^11^1+\begin{pmatrix}2\\2\end{pmatrix}0^21^0=0^0$

The binomial series, on the other hand, is a more general concept where $n$ can be any real number, including negative and fractional values. If one of the variables in eq319 is equal to one and the absolute value of the other variable is less than one, the binomial series provides a way to express an infinite series that converges:

$(1+x)^n=\sum_{k=0}^{\infty}\begin{pmatrix}n\\k\end{pmatrix}x^k\;\;\;\;\;\vert x\vert < 1$

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August 29, 2024

Generating function for the Legendre Polynomials

The generating function for the Legendre polynomials is a mathematical tool that, when expanded as a power series, produces Legendre polynomials as its coefficients in terms of a variable.

Legendre polynomials often arise in problems involving spherical harmonics. An example (see diagram above) is the multipole expansion $\frac{1}{\vert\boldsymbol{\mathit{r}}_i-\boldsymbol{\mathit{r}}_j\vert}=\frac{1}{r_i}\sum_{n=0}^{\infty}\biggr$\frac{r_j}{r_i}\biggr$^lP_l(cos\theta)$ . As the two points $q_i,q_j$ and the angle $\theta$ between them form a triangle, the Legendre polynomials $P_l(cos\theta)$ are related to the cosine rule $c=\sqrt{a^2+b^2-2abcos\theta}$ , which can be rearranged to

$\frac{b}{c}=\frac{1}{\sqrt{1-2tx+t^2}}\;\;\;\;\;\;\;\;343$

where $t=\frac{a}{b}$ and $x=cos\theta$ .

Since $\vert b\vert>\vert a\vert$ , we have $\vert t\vert<1$ and hence $\vert 2tx-t^2\vert<1$ . This implies that we can expand the RHS of eq343 as a binomial series:

$\frac{1}{\sqrt{1-2tx+t^2}}=\boldsymbol{1}+\boldsymbol{x}t+\biggr(\boldsymbol{\frac{3}{2}x^2-\frac{1}{2}}\biggr)t^2+\biggr(\boldsymbol{\frac{5}{2}x^3-\frac{3}{2}x}\biggr)t^3+\cdots\;\;\;\;\;\;\;\;344$

The coefficients of $t$ are the Legendre polynomials. Therefore, the generating function $G(x,t)$ for the Legendre polynomials $P_l(x)$ is

$G(x,t)=\frac{1}{\sqrt{1-2tx+t^2}}=\sum_{l=0}^{\infty}P_l(x)t^l\;\;\;\;\;\;\;\;345$

Next article: Recurrence relations of the Legendre polynomials

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